sqrt(2) modulo primes

As everyone learns in infancy, 2 is a square modulo any prime of the
form 8n+1 or 8n-1. Actually 2 is a fourth power modulo any prime 8n-1,
and its fourth roots are explicitly 2^n and -2^n. (There are no
others.) But what about sqrt(2) mod primes 8n+1? Any formula known? For
comparison, if 4n+1 is prime then the sqrt's of -1 mod 4n+1 are (2n)!
and its negative.

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